PeriodicLongitudinalTTSubspace5
plain-language theorem explainer
Concrete membership predicate for transverse-traceless edge perturbations on the fixed 5-periodic torus under the longitudinal gauge: an edge strain is TT when it is orthogonal to the image of the finite longitudinal gauge map built from vertex-vector delta generators. Gravity Track 1.D cites it whenever Regge TT Hessian and lattice Lichnerowicz bilinears are compared on TT modes. It is a pure abbreviation specializing the orthogonal-to-gauge construction to that gauge map.
Claim. Write $\mathrm{TT}_{\mathrm{long}}$ for the set of periodic edge perturbations $\varepsilon$ orthogonal to the image of the longitudinal gauge map $G$, where $G$ sends coefficient functions on (periodic vertex) $\times\{0,1,2\}$ to edge strains via vertex-vector Kronecker-delta generators. Then $\varepsilon\in\mathrm{TT}_{\mathrm{long}}$ means $\varepsilon$ is orthogonal to $G(c)$ for every coefficient function $c$.
background
Track 1.D opens the tensor/shear sector because the Track 1.B conformal ansatz (one scalar per vertex, edge strains by averaging endpoints) cannot represent pure shear and therefore cannot cover transverse-traceless weak-field modes. This module separates independent edge perturbations from vertex-conformal ones and builds a concrete periodic TT slice.
The longitudinal gauge index set is one spatial component at one periodic vertex (vertex $\times$ three directions). The associated gauge map sends real coefficients on that index set to periodic edge perturbations by the finite generator map whose basis elements are vertex-vector deltas (Kronecker delta on the finite index). The TT predicate is the orthogonal complement, in the periodic edge inner product, to the image of that map.
Upstream, the Kronecker delta supplies the generator coefficients; the gauge index and map abbreviations fix the concrete finite basis used throughout the Hessian/Lichnerowicz matching lemmas.
proof idea
Definitional abbreviation only: instantiate the generic orthogonal-to-gauge TT predicate at the longitudinal coefficient type (functions on the longitudinal gauge index) and at the concrete longitudinal gauge map. No tactic proof; the body is the specialization of that orthogonal construction.
why it matters
This predicate is the TT hypothesis threaded through the Track 1.D operator-matching stack. Downstream bilinear theorems state that once Regge TT Hessian and lattice Lichnerowicz agree as operators on TT modes, their bilinears agree whenever the right input lies in this subspace (match-data, origin-column, translated, and relative-translated coefficient certificates). The master handoff endpoint packages the same condition: bilinear and quadratic TT energy matching after pointwise operator identification on TT modes.
It also appears in the relative-frame orthogonality obligation (every TT strain orthogonal to row-frame translates of the combined conformal/longitudinal generator map) and in kernel residual/row data for the TT Hessian. In the gravity track this is the discrete stand-in for the continuum TT gauge slice needed before shear/GW content can be compared to continuum Lichnerowicz theory. It does not itself close continuum identification; it fixes the finite longitudinal-gauge TT cut used by those reductions.
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